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Sg                          ANALYTICAL MECHANICS
4. A wheel of radius a and weight IF stands on rough horizontal ground. If ju is the coefficient of friction between the whorl and (he ground'find the smallest weight which must ho suspended at one end of the horizontal diameter in order to move the wheel.
1.   A table of negligible weight has three logs, the feet forming an equilateral triangle.   Find the proportion of the weight carried by the legs when a particle is placed on the table.
2.   A rectangular board is supported in a vertical position hy two smooth pegs in a vertical wall.    Show that if one of the diagonals is parallel to the line joining the pegs the other diagonal Ls vortical.
3.   A uniform rod rests with its two ends on .smooth inclined planes making angles a and /? with the horizon.   Where must a weight equal to that of the rod be clamped in order that the rod may rest- horizontally?
4.   A uniform ladder rests against a rough vertical wall.    Show that the least angle it can make with the horizontal floor on which it rests is
given by tan 6 =  ~-^L, where /JL and // are the eoeflieionts of friction 2 /z
for the floor and the wall, respectively.
5.   A uniform rod is suspended by two equal .strings attached to the >ends.   In position of equilibrium the strings art* parallel and the lar in horizontal.   Find the torque which will turn the bar, about a vertical axis, through an angle 6 arid keep it in equilibrium at. that position.
6.   The line of hinges of a door makes an angle.  with the vertical Find the resultant torque when the door makes an angle fi with its equilibrium position.
7.   The lines of action of four forces form a quadrilateral.    If the magnitude of the forces are a, 6, c, d times the Hides of flic quadrilateral find the conditions of equilibrium.
8.   A force acts at the middle point of each suit* of a plane polygon. Each force is proportional to the length of the side* it actn upon nud is perpendicular to it.   Prove that the polygon will be in equilibrium If all the forces are directed towards the inside of the polygon.
9.   A force acts at each vertex of a piano convex polygon in it <l5w-tion parallel to one of the sides forming the vortex.   Show tlmt if tlw forces^are proportional to the sides to which they are parallel ami if their directions are in a cyclic order their resultant itTa couple.
10.   A uniform chain of length I hangs over a rough horizontal eylimler of radius a.   Find the length of the portions which hang vertically \vlmn